Why Does Distance From Axis Affect Moment of Inertia?

In summary, the concept of moment of inertia is similar to mass in linear dynamics, as it offers resistance in rotational dynamics. It describes an object's resistance to changes in its rotational motion. This resistance is affected by both the object's mass and how far the force is applied from its axis of rotation. The further the distance, the less resistance, and the closer the distance, the more resistance. However, the relationship between moment of inertia and distance is not linear, as it is proportional to r^2, not just r. This can be better understood by considering the torque applied with a given force, and how it affects the rotational inertia of the object.
  • #1
ninja319
2
0
this might sound like a strange and general question but I am a bit confused so please help me, i was wondering what exactly moment of inertia is

- we know just like mass offers resistance in linear dynamics, we need something that offers a similar kind of resistance in rotational dynamics
- i imagined a flat, somewhat heavy, metal plate (lets assume completely cylindrical and a relatively small thickness compared to the radius. more like a flat metallic pizza) floating in empty space. it has a chunk (a slice) cut away from it (so it looks like a metallic pizza with one slice missing).
- now i have to find what is causing the resistance when I am trying to apply force and rotate it.
- i know of the plate causes rotation . so 'I' is proportional to 'm'.
- now a similar amount of force is applied on a point closer to the axis of rotation and a point further away from it. the force applied further away (from observation) shows less resistance. that is it is harder to change the angular velocity when you apply the force closer to a point on the axis of rotation. so further the distance we move from the axis of rotation (more the r) lesser the resistance becomes. therefore, 'I' is inversely proportional to r.
- but that is clearly not the case as I = mr^2 not I = m/r
- I am sure i went wrong in the reasoning. can someone please tell me what it is?
 
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  • #2
sorry, correction

"i know of the plate causes rotation . so 'I' is proportional to 'm'."

- i know that more the mass of the plate, more the resistance. so 'I' is proportional to 'm'
 
  • #3
ninja319 said:
- now a similar amount of force is applied on a point closer to the axis of rotation and a point further away from it. the force applied further away (from observation) shows less resistance. that is it is harder to change the angular velocity when you apply the force closer to a point on the axis of rotation. so further the distance we move from the axis of rotation (more the r) lesser the resistance becomes. therefore, 'I' is inversely proportional to r.
- but that is clearly not the case as I = mr^2 not I = m/r
But the amount of angular acceleration is proportional to how far from the axis the force is applied. Since you are not changing the mass distribution of your object, its rotational inertia remains the same. But the torque you are applying with a given force is proportional to r.

In your example you are changing the torque, not the rotational inertia. (Analogous to changing the force, but not the mass, and getting a different linear acceleration.)
 
  • #4
Last edited by a moderator:

What is moment of inertia?

Moment of inertia is a physical property of a rigid body that measures its resistance to rotational motion around a specific axis. It is also known as rotational inertia or angular mass.

How is moment of inertia calculated?

The moment of inertia of an object can be calculated by multiplying the mass of the object by the square of its distance from the axis of rotation. It is also affected by the shape and distribution of mass in the object.

Why is moment of inertia important?

Moment of inertia is important because it helps us understand and predict how objects will behave when rotating. It is also a crucial factor in the design of machines and structures that involve rotational motion.

What are some real-life examples of moment of inertia?

Some examples of moment of inertia in everyday life include the spinning of a top, the rotation of a bicycle wheel, and the swinging of a pendulum.

How does moment of inertia differ from mass?

Moment of inertia and mass are two different physical properties of an object. Mass measures the amount of matter in an object, while moment of inertia measures its resistance to rotational motion. They are related, but not interchangeable, as an object's mass does not necessarily determine its moment of inertia.

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